Numerical Evidence for a Haagerup Conformal Field Theory

Abstract

We numerically study an anyon chain based on the Haagerup fusion category, and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is ∼ 2. Fusion categories generalize the concept of finite group symmetries to non-invertible symmetry operations, and the Haagerup fusion category is the simplest one which comes neither from finite groups nor affine Lie algebras. As such, ours is the first example of conformal field theories which have truly exotic generalized symmetries.

Non-invertible symmetries of $$ \mathcal{N} $$ = 4 SYM and twisted compactification

“…However, as we will now see, at different values of τ YM the theories do possess intrinsically non-invertible defects, which can give rise to constraints going beyond those obtainable from invertible symmetries. 11 To obtain a non-trivial duality symmetry, it was crucial that we included an {S, T} transformation, and furthermore that we were able to set τ YM to a fixed point of this actions. Restricting ourselves to the fundamental domain, the only other fixed points are then τ YM = i∞ and τ YM = e 2πi/3 .…”

“…This leads to the notion of "generalized symmetries", and includes higher-form symmetries (for which the topological operator has codimension greater than one) and non-invertible symmetries (for which the associated topological operators do not possess an inverse, and as such do not form a group). The latter will be especially important for our purposes -they have been studied in two-dimensions in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and more recently in higher dimensions in [18][19][20][21][22][23][24][25][26]. Despite going beyond our traditional notions of symmetry, the topological nature of these operators suggests that they give rise to constraints similar to those provided by ordinary symmetries, including constraints on RG flows [4, 6-8, 21, 26].…”

Critical Lattice Model for a Haagerup Conformal Field Theory

“…Near the completion of this work, we learned that a critical anyonic chain Hamiltonian for the Haagerup fusion category H 3 was obtained independently [61]. Their numerical evidence also indicates a central charge c ¼ 2 CFT for this Hamiltonian.…”

Symmetry TFTs for Non-Invertible Defects

“…To reproduce the fusion rules of duality defects (5.14) from the twist defects, let us first consider the collision of the twist defect with the Dirichlet boundary condition. 18 The Dirichlet boundary condition of the dynamical field b of the SymTFT means that b in S (1,0) (τ ) = e 2πi/N τ b is valued in relative cohomology H 2 (X 5 , X 4 , Z N ). To integrate such b on a two-cycle τ , the twocycle τ should be in relative homology H 2 (X 5 , X 4 , Z N ).…”

“…Non-invertible symmetries have been the subject of an extensive literature in (1 + 1)d (see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]), but have only recently been generalized to spacetime dimensions greater than two . The constructions of non-invertible symmetries in higher dimensions that have appeared in the literature so far involve the following techniques,…”